Holographic Wilson Loops and Topological Insulators Andy OBannon - - PowerPoint PPT Presentation
Holographic Wilson Loops and Topological Insulators Andy OBannon Crete Center for Theoretical Physics February 21, 2012 Credits Work in progress with: John Estes Instituut voor Theoretische Fysica Katholieke Universiteit Leuven
Holographic Wilson Loops and Topological Insulators
Andy O’Bannon
Crete Center for Theoretical Physics February 21, 2012
Credits
John Estes
Work in progress with:
Timm Wrase Efstratios Tsatis
Instituut voor Theoretische Fysica Katholieke Universiteit Leuven Cornell University University of Patras
Outline:
Topological Insulators topological quantum number distinct from vacuum U(1) symmetry mass gap in charged sector
Topological Insulators A number invariant under continuous deformations that: topological quantum number
Topological Insulators Cannot be classified (just) by local order parameter topological quantum number
Example: Integer QHE
σxy = ne2 h
z Bhematic illustration of quantum Hall plateau t
σxx = 0
On plateaux:
topological quantum number
Z
Example: Integer QHE
Edge modes: chiral fermions
IQHE state n=1 Vacuum n=0 Observed via: Transport Directly (ARPES)
n = # of edge modes
Effective Description
S = k 4π
Example: Integer QHE
Ji = δS δAi = k 2π ǫijEj
quantization of quantization of k
=
σxy
T
d CdTe HgTe CdTe
(a)
(2+1)d (3+1)d Bi2Se3
Bi2Te3 Sb2Te3
Fu and Kane 2007 Bernevig, Hughes, Zhang 2006 König et al. 2007 Hsieh et al. 2008
(3+1)d T
UV: some band structure
momentum energy band gap conduction band valence band
T : M → M ∗
M ∈ R
(3+1)d T
IR: non-interacting electrons
M > 0 M < 0
S =
ψ (i∂ − A − M) ψ
Extreme IR: axion electrodynamics
θ =
π M < 0
(3+1)d T
L = 1 8π
E2 − 1 µ
θ 4π2 E · B
Z2 topological quantum number
mod 2π
(3+1)d T
θ = 0 θ = π
θ(z)
z
z = 0
Edge modes: (2+1)d Dirac fermions
ARPES
I(ω, k) ∝ f(ω)R(ω, k)
Angle Resolved Photo-Emission Spectroscopy
(3+1)d T
0.0 0.1 0.2
0.1 0.2
ky (Å )
kx (Å )
(b)
Bi2Se3
Xia et al. 0812.2078 Hsieh et al. Nature 460, 1101
Edge modes: (2+1)d Dirac fermions
Jumping θ-angle: image dyons
at level of free Dirac Hamiltonians
CLASSIFICATION
Choices:
class\d 1 2 3 4 5 6 7 T C S A Z Z Z Z AIII Z Z Z Z 1 AI Z 2Z Z2 Z2 + BDI Z2 Z 2Z Z2 + + 1 D Z2 Z2 Z 2Z + DIII Z2 Z2 Z 2Z − + 1 AII 2Z Z2 Z2 Z − CII 2Z Z2 Z2 Z − − 1 C 2Z Z2 Z2 Z − CI 2Z Z2 Z2 Z + − 1
Periodic Table of TI’s
Schnyder, Ryu, Furusaki, Ludwig 2008 Kitaev 2009
class\d 1 2 3 4 5 6 7 T C S A Z Z Z Z AIII Z Z Z Z 1 AI Z 2Z Z2 Z2 + BDI Z2 Z 2Z Z2 + + 1 D Z2 Z2 Z 2Z + DIII Z2 Z2 Z 2Z − + 1 AII 2Z Z2 Z2 Z − CII 2Z Z2 Z2 Z − − 1 C 2Z Z2 Z2 Z − CI 2Z Z2 Z2 Z + − 1
Integer QHE Periodic Table of TI’s
class\d 1 2 3 4 5 6 7 T C S A Z Z Z Z AIII Z Z Z Z 1 AI Z 2Z Z2 Z2 + BDI Z2 Z 2Z Z2 + + 1 D Z2 Z2 Z 2Z + DIII Z2 Z2 Z 2Z − + 1 AII 2Z Z2 Z2 Z − CII 2Z Z2 Z2 Z − − 1 C 2Z Z2 Z2 Z − CI 2Z Z2 Z2 Z + − 1
Time-reversal invariant Periodic Table of TI’s
What can happen with
electrons?
No description in terms of non-interacting electrons Example: Fractional QHE
σxy = p q e2 h
Interacting electrons Fractional QHE Integer QHE Non-interacting electrons T
Interacting electrons Fractional QHE Integer QHE Non-interacting electrons T
Fractional T
Similar to QCD?
Wen 1999 Maciejko, Qi, Karch, Zhang 2010, 2011
Fractionalization
Swingle, Barkeshli, McGreevy, Senthil 2010
Fractionalization Electron Charge -1 under Singlet under
U(1)EM
G
“Statistical Gauge Fields”
U(1)EM → U(1)EM × G
Fractionalization
Electron Baryon Fractional Electron Quark
Statistical Gauge Fields
Gluons
Similar to QCD?
= = =
Fractionalized phase Non-confining phase
=
U(1)EM × G ≃ U(1)B × SU(Nc)
with jumping θ-angle (3+1)d Fractional T
N = 4 supersymmetric SU(Nc) Yang-Mills
Karch 2009
GOAL
Compute the potential between test charges
N = 4 SYM with jumping θ-angle
in
Outline:
N = 4 SUSY SU(Nc)
(3+1)d YM
− 1 2g2 tr(DµΦiDµΦi) + 1 4g2 tr([Φi, Φj][Φi, Φj])
L = − 1 4g2 trFµνF µν + θ 32π2 ǫµνρσtrFµνFρσ
SO(4, 2) × SO(6)
λ = g2Nc
βλ = 0
N = 4 SUSY SU(Nc)
(3+1)d YM
λ = g2Nc
Nc → ∞
fixed
− 1 2g2 tr(DµΦiDµΦi) + 1 4g2 tr([Φi, Φj][Φi, Φj])
L = − 1 4g2 trFµνF µν + θ 32π2 ǫµνρσtrFµνFρσ
=
SO(4, 2) × SO(6)
=
N = 4 SYM
Nc, λ ≫ 1
IIB SUGRA
AdS5 × S5
global symmetry
isometry
“holographic”
r = ∞
r = 0
boundary
ds2 = dr2 r2 + r2 −dt2 + dx2 + dy2 + dz2
The Janus Solution
Bak, Gutperle, Hirano 2003
Solution of type IIB SUGRA
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
The Janus Solution
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
One-parameter dilatonic deformation of AdS5 × S5
AdS4 slicing of AdS5
h(x) = 1 1 − x2
ds2 = R2 h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
isometry manifest
SO(3, 2) × SO(6)
z < 0 z = 0 z > 0
AdS4 slicing of AdS5
z < 0 z = 0 z > 0
x
AdS4
The Janus Solution
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
One-parameter dilatonic deformation of AdS5 × S5
The Janus Solution
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
Isometry SO(3, 2) × SO(6)
The Janus Solution
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
One free parameter
γ ∈ (3/4, 1)
The Janus Solution
ds2 = R2 γ−1h(x)2dx2 + h(x)ds2
AdS4 + ds2 S5
h(x) = γ
4γ − 3 ℘(x) + 1 − 2γ
℘(χ)
σ(x − χ) − 2ζ(χ)x
(and the five-form)
Breaks ALL SUSY
e2φ−
e2φ+
The Janus Solution Free parameter: jump in dilaton
“r = ∞”
“r = 0”
x
Roman god of beginnings, transitions, gates, and doorways Towards Past Towards Future Janus
Root of “January” and “Janitor” Towards Past Towards Future Janus
− 1 2g2 tr(DµΦiDµΦi) + 1 4g2 tr([Φi, Φj][Φi, Φj])
L = − 1 4g2 trFµνF µν + θ 32π2 ǫµνρσtrFµνFρσ
g−
g+
z
z = 0
g(z)
e2φ = g2/2π
− 1 2g2 tr(DµΦiDµΦi) + 1 4g2 tr([Φi, Φj][Φi, Φj])
L = − 1 4g2 trFµνF µν + θ 32π2 ǫµνρσtrFµνFρσ
g−
g+
z
z = 0
g(z)
Jumping coupling
− 1 2g2 tr(DµΦiDµΦi) + 1 4g2 tr([Φi, Φj][Φi, Φj])
L = − 1 4g2 trFµνF µν + θ 32π2 ǫµνρσtrFµνFρσ
Jumping coupling Preserves SO(3,2) x SO(6) Breaks all SUSY
“Conformal Interface”
Dielectric Interfaces
Image charges!
L = 1 8π
E2 − 1 µ
Jumping dilaton Jumping axion
τ → aτ + b cτ + d
ab − cd = 1
a, b, c, d ∈ R
τ = C0 + ie−2φ
SL(2, R)
Jumping coupling
Topological Insulator
Jumping θ-angle
τ → aτ + b cτ + d
ab − cd = 1
a, b, c, d ∈ R
τ = θ 2π + i2π g2
SL(2, R)
Outline:
Wilson Loops in N = 4 SYM
R = Nc
SU(Nc)
T
WR[C] = 1 Nc trRP exp
ds
xµ + Φiθi| ˙ x|
Wilson Loops in N = 4 SYM
R = Nc
SU(Nc)
T
L
WR[C] = 1 Nc trRP exp
ds
xµ + Φiθi| ˙ x|
Wilson Loops in N = 4 SYM
WR[C] = 1 Nc trRP exp
ds
xµ + Φiθi| ˙ x|
L
V (L) = − lim
T →∞
1 T lnW[C]
Wilson Loops in N = 4 SYM Perturbatively
λ ≪ 1
− | ˙ x(s)|| ˙ x(˜ s)|θiθjPΦi(x(s))Φj(x(˜ s))
W[C] = 1 − Nc 2
ds
d˜ s ( ˙ xµ(s) ˙ xν(˜ s)PAµ(x(s))Aν(x(s))
Wilson Loops in N = 4 SYM Sum “ladder” diagrams
+ + · · ·
Wilson Loops in N = 4 SYM Holographically
λ ≫ 1
b.
r = ∞
r = 0
C
Maldacena Rey and Yee 1998
Wilson Loops in N = 4 SYM
V (L) = − lim
T →∞
1 T lnW[C] = lim
T →∞
A T
SNG|solution = A
W[C] ∝ e−A
SNG = − 1 2πα′
Wilson Loops in N = 4 SYM
SNG|solution
diverges at
r = ∞
Infinite self-energy
b.
C
Wilson Loops in N = 4 SYM Legendre transform
Drukker, Gross, Ooguri 1999
A = SNG −
Wilson Loops in N = 4 SYM
“Straight string” Legendre transform cancels the divergence!
A = 0
W[C] = 1
due to SUSY
⇒
N = 4 SYM jumping g
Subtracting straight string Subtracting self-energy
=
b.
C
Wilson Loops from AdS5
Legendre transform =
Holography Ladder Diagrams
V (L) = − 1
4π 2λ L ,
λ ≪ 1 − 1
π √ 2λ L ,
λ ≫ 1.
λ ≫ 1
Maldacena Rey and Yee 1998 Erickson Semenoff Zarembo 2000V (L) = − 4π2 Γ (1/4)4 √ 2λ L
Conformal Interface Wilson Loops in N = 4 SYM
X 3L D
X 3 X X 3 3 right leftz z
zright
zleft
V (L, D) = f(λ, D/L) L
X 3L D
z
Conformal Interface Wilson Loops in N = 4 SYM
PAµ(x(s))Aν(x(s)) acquires image terms
PΦi(x(s))Φj(x(˜ s)
unchanged
Clark, Freedman, Karch, Schnabl 2004
Perturbatively
Wilson Loops in N = 4 SYM
Holographically
Conformal Interface
Outline:
gI
gII
˜ Q
˜ Q = Qg2
II − g2 I
g2
II + g2 I
Q
Electromagnetism
−z
+z
gI
gII
attracted to side with SMALLER coupling
Q
˜ Q
Q
Electromagnetism
−z
+z
Electromagnetism
V g2
I/4π
g2
I > g2 II
Q = +1
g2
II
g2
I
z
gI
gII
W[C]
C
with straight-line
˜ Q
Q
−z
+z
N = 4 SYM jumping g
N = 4 SYM jumping g
A = 0
(!)
Straight string in Janus
A = SNG −
V = lim
T →∞
A T = 0
Interaction energy with image charge
N = 4 SYM jumping g
z
g2 e2g2
V
gI
gII
˜ Q = Qg2
II − g2 I
g2
II + g2 I
Q1 = +1
Q2 = −1
˜ Q1
˜ Q2
D D
L
Electromagnetism
−z
+z
gI
gII
Q1 = +1
Q2 = −1
˜ Q1
˜ Q2
D D
L
Electromagnetism
−z
+z
V = g2
I4π
L + Q1 ˜ Q1 2D + Q2 ˜ Q2 2D + Q1 ˜ Q2 √ L2 + 4D2
Electromagnetism
gI
gII
Q1 = +1
Q2 = −1
˜ Q1
˜ Q2
D D
L
Again attracted to side with SMALLER coupling
−z
+z
Electromagnetism
2 1 1 2 5 3 1 1 3g2
II = 1
2g2
I
g2
I
V L g2
I/4πz L
Electromagnetism
V L g2
I/4π − (images)
10 5 5 10 2.0 1.5 1.0 0.5 0.5 1.0g2
II = 1
2g2
I
g2
I
z L
gI
gII
Q1 = +1
Q2 = −1
˜ Q1
˜ Q2
D D
L
−z
+z
N = 4 SYM jumping g
W[C]
C
with rectangular
N = 4 SYM jumping g
g2 e2g2
z L
V L
N = 4 SYM jumping g
4 2 2 4 2.2 2.0 1.8 1.6 1.4 1.2 1.0g2 e2g2
z L
V L
− (images)
Q
Electromagnetism
θII
θI
( ˜ Qe, ˜ Qm)
˜ Qe = −Q g4 (θII − θI)2 16π2 + g4 (θII − θI)2 ˜ Qm = −4πQ g2(θII − θI) 16π2 + g4 (θII − θI)2
−z
+z
Q
Electromagnetism
( ˜ Qe, ˜ Qm)
˜ Qe = −Q g4 (θII − θI)2 16π2 + g4 (θII − θI)2 ˜ Qm = −4πQ g2(θII − θI) 16π2 + g4 (θII − θI)2
θII
θI
−z
+z
Q
Electromagnetism
( ˜ Qe, ˜ Qm)
Interface always attractive!
θII
θI
−z
+z
Electromagnetism
z
V g2/4π
Q
θII
θI
( ˜ Qe, ˜ Qm)
˜ Qe = −Q g4 (θII − θI)2 16π2 + g4 (θII − θI)2 ˜ Qm = −4πQ g2(θII − θI) 16π2 + g4 (θII − θI)2
N = 4 SYM jumping θ
−z
+z
Q
θII
θI
( ˜ Qe, ˜ Qm)
˜ Qe = −Q g4 (θII − θI)2 16π2 + g4 (θII − θI)2 ˜ Qm = −4πQ g2(θII − θI) 16π2 + g4 (θII − θI)2
N = 4 SYM jumping θ
−z
+z
N = 4 SYM jumping θ
z
V √ 2λ/2π
Electromagnetism
Interface always attractive!
θII
θI
Q1 = +1
Q2 = −1
D D
L
( ˜ Qe
1, ˜
Qm
1 )
( ˜ Qe
2, ˜
Qm
2 )
−z
+z
Electromagnetism
1 0.5 0.5 1 30 20 10V L g2/4π
z L
g2∆θ = 102
Electromagnetism
z L
V L g2/4π − (images)
15 10 5 5 10 15 1 0.8 0.6 0.4 0.2g2∆θ = 102
θII
θI
Q1 = +1
Q2 = −1
D D
L
( ˜ Qe
1, ˜
Qm
1 )
( ˜ Qe
2, ˜
Qm
2 )
˜ Qe = −Q g4 (θII − θI)2 16π2 + g4 (θII − θI)2 ˜ Qm = −4πQ g2(θII − θI) 16π2 + g4 (θII − θI)2
N = 4 SYM jumping θ
−z
+z
N = 4 SYM jumping θ
z L
θ = 0
V L √ 2λ
θ = π
θ = 0
θ = π
N = 4 SYM jumping θ
4 2 2 4 1.430 1.425 1.420 1.415 1.410 1.405z L
V L √ 2λ/2π − (images)
θ = 0
θ = π
0.2 0.1 0.1 0.2 1.4068 1.4066 1.4064 1.4062 1.4060 1.4058 1.4056N = 4 SYM jumping θ
z L
V L √ 2λ/2π − (images)
∆θ = 20π
N = 4 SYM jumping θ
Outline:
Future Directions